Abstract
In this paper, we present recent results in harmonic analysis in the real line and in the half-line , which show a closed relation between Hermite and Laguerre functions, respectively, their symmetry groups and Fourier analysis. This can be done in terms of a unified framework based on the use of rigged Hilbert spaces. We find a relation between the universal enveloping algebra of the symmetry groups with the fractional Fourier transform. The results obtained are relevant in quantum mechanics as well as in signal processing as Fourier analysis has a close relation with signal filters. In addition, we introduce some new results concerning a discretized Fourier transform on the circle. We introduce new functions on the circle constructed with the use of Hermite functions with interesting properties under Fourier transformations.
Highlights
The seminal work by Fourier of 1807, published in 1822 [1], about the solution of the heat equation had a deep impact in physics and mathematics as is well known
In recent works [5,6], we started an attempt to reformulate the harmonic analysis on the real line to obtain a global description of the Hermite functions, the Weyl–Heisenberg Lie algebra and the Fourier analysis in the framework of rigged Hilbert spaces (RHS) that we present here in a more formal way
The fundamental message of the present paper is to show how a class of different and apparently unrelated mathematical objects, such as classical orthogonal polynomials, Lie algebras, Fourier analysis, continuous and discrete bases and RHS, can be fully wived as a branch of harmonic analysis, with applications in quantum mechanics and signal processing, among other possible applications
Summary
The seminal work by Fourier of 1807, published in 1822 [1], about the solution of the heat equation had a deep impact in physics and mathematics as is well known. In recent works [5,6], we started an attempt to reformulate the harmonic analysis on the real line to obtain a global description of the Hermite functions, the Weyl–Heisenberg Lie algebra and the Fourier analysis in the framework of rigged Hilbert spaces (RHS) that we present here in a more formal way. The Fourier transform relates two continuous bases which are used in the description of one-dimensional quantum systems on the whole real line These are the coordinate and momentum representations, naturally connected with the position and the momentum operator [7,8], respectively. To understand the importance of the present research, let us remark that Hermite and Laguerre functions are bases of spaces of square integrable functions, no matter whether real or complex, defined on R and R+ , respectively. We would like to add that this discussion may be related with some integral transforms of the type Fourier-like, Laplace-like or Sumudu-like transforms [32,33,34,35,36]
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